Probabilistic approaches to rational points on algebraic surfaces
Appears in collection : AGCT - Arithmetic, Geometry, Cryptography and Coding Theory / AGCT - Arithmétique, géométrie, cryptographie et théorie des codes 2023
The Brauer group of a del Pezzo or a K3 surface over a number field is thought to govern the existence of rational points. A large piece of this group is determined by the Galois-module structure on the geometric Picard group of a surface. I will present work in progress that, given equations for a low-degree del Pezzo or K3 surface, determines its algebraic Brauer group with a high degree of confidence. I will also indicate how e˙ective versions of the Chebotarev density can certify probabilistic results, under GRH. Technology permitting, I will show a live demo.N.B. This is joint work with Austen James.
